Problem: $ B = \left[\begin{array}{r}2 \\ 2 \\ 2\end{array}\right]$ $ D = \left[\begin{array}{r}3 \\ 0 \\ 3\end{array}\right]$ Is $ B D$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ B$ , have? How many rows does the second matrix, $ D$ , have? Since $ B$ has a different number of columns (1) than $ D$ has rows (3), $ B D$ is not defined.